Primary extension
In field theory, a branch of algebra, a primary extension L of K is a field extension such that the algebraic closure of K in L is purely inseparable over K.[1]
Properties
- An extension L/K is primary if and only if it is linearly disjoint from the separable closure of K over K.[1]
- A subextension of a primary extension is primary.[1]
- A primary extension of a primary extension is primary (transitivity).[1]
- Any extension of a separably closed field is primary.[1]
- An extension is regular if and only if it is separable and primary.[1]
- A primary extension of a perfect field is regular.
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References
- Fried & Jarden (2008) p.44
- Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. pp. 38–44. ISBN 978-3-540-77269-9. Zbl 1145.12001.
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